22nd IFIP TC 7

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22nd IFIP TC 7 Conference on System Modeling
and Optimization Analysis of the convective
instability of the two-dimensional wake
D.Tordella , S.Scarsoglio and M.Belan
Dipartimento di Ingegneria Aeronautica e
Spaziale, Politecnico di Torino
Dipartimento di Ingegneria Aeronautica e
Spaziale, Politecnico di Milano Turin,
Italy, July 18-22, 2005
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Introduction A linear stability study is here
presented for two dimensional non-parallel flows
in the intermediate and far wake behind a
circular body. The hydrodynamic stability
analysis is developed within the linear theory of
normal modes through a perturbative approach, it
is observed the behavior of small oscillations
applied to the base flow. An analytic expression
of the base flow according to Navier-Stokes model
is given by an asymptotic expansion (Tordella and
Belan, 2003 Belan and Tordella, 2002), which
considers non-parallelism effects (such as
exchange of transverse momentum and
entrainment). It is supposed that the system
slowly evolves in space (Tordella and Belan,
2005) and also in time using multiple spatial
and temporal scales, we can verify how this
evolution influences the stability
characteristics and discuss about a validity
domain for parallel flow.
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Basic equations and physical problem Steady,
incompressible and viscous base flow described by
continuity and Navier-Stokes equations with
dimensionless quantities U(x,y), V(x,y), P(x,y)
and ? ?cost
R ?UcD/?
Boundary conditions symmetry to x, uniformity at
infinity and field information in the
intermediate wake
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To analytically define base flow, its domain is
divided into two regions both described by
Navier-Stokes model
Inner region flow -gt

, Outer region
flow -gt
,
Physical quantities involved in matching criteria
are the pressure longitudinal gradient, the
vorticity and transverse velocity. Inner and
outer expansions are used to obtain the composite
expansion
which is, by construction, continuous and
differentiable over the whole domain. Accurate
representation of the velocity and pressure
distributions (obtained without restrictive
hypothesis) and analytical simplicity of
expansions. Here we take the inner expansion up
to third order as base flow solution for the wake.
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vc
vo
vi
ui
uo
uc
po - p?
pc - p?
R 34, x/D 20. Fourth order of accuracy
Inner, outer and composite expansions for
velocity and pressure.
pi - p?
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R 34, x/D 20. Comparison of the present
fourth order outer and composite, Chang's outer
and composite (1961), Kovasznay's experimental
(1948) and Berrone's numerical (2001)
longitudinal velocity distributions.
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Stability theory Base flow is excited with small
oscillations. Perturbed system is described by
Navier-Stokes model Subtracting base flow
equations from those concerning perturbed flow
and neglecting non linear oscillating terms, the
linearized perturbative equation in term of
stream function is
Normal modes theory Perturbation is
considered as sum of normal modes, which can be
treated separately since the system is linear.
complex eigenfunction,
u(x,y,t) U(x,y) u(x,y,t) v(x,y,t) V(x,y)
v(x,y,t) p(x,y,t) P0 p(x,y,t)
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• 
  
  k0 wave number
  
• h0 k0 i s0 complex wave
  number s0 spatial growth rate
• ?0 ?0 i r0
  complex frequency ?0
  frequency
• 
  
  r0 temporal growth rate
• Perturbation amplitude is proportional to
  
• r0 0 for at least one mode
  unstable flow
• r0 0 for all modes
  stable flow
• s0 0 for at least one mode
  convectively unstable flow
• s0 0 for all modes
  convectively stable flow
• Convective instability r0 0 for all modes, s0
  0 for at least one mode. Perturbation spatially
  amplified in a system moving with phase velocity
  of the wave but exponentially damped in time at
  fixed point.
• Absolute instability r0 0 (vg??0/?k00
  local energy increase) for at least one mode.
  Temporal amplification of the oscillation at
  fixed point.

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• Stability analysis through multiscale approach
• Slow spatial and temporal evolution of the system
  slow variables x1 ?x, t1 ?x.
• 1/R is a dimensionless parameter that
  characterizes non-parallelism of base flow.
• Hypothesis and
  are expansions in term of ?
• 
  
• By substituting in the linearized perturbative
  equation, one has
• (ODE dependent on ) ?(ODE dependent on
  , ) O (?2)
• Order zero theory. Homogeneous Orr-Sommerfeld
  equation (parametric in x1 ).
• where
  , and A(x1,t1) is the slow
  spatio-temporal modulation, determined at next
  order.
• By numerical solution
  eigenfunctions ?0 and a discrete set of
  eigenvalues ?0n

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First order theory. Non homogeneous
Orr-Sommerfeld equation (x1 parameter).
is related to base flow and consider
non-parallel effects through transverse velocity
presence
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To obtain first order solution, the non
homogeneous term is requested to be orthogonal to
every solution of the homogeneous adjoint
problem, so that Keeping in mind that

, the complete problem gives First order
corrections h1 e ?1 are obtained by resolving
numerically the evolution equation for modulation
and differentiating numerically a(x,t) with
respect to slow variables.
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Perturbative hypothesis Saddle points
sequence From order zero theory its possible
having a first approximation of the dispersion
relation ?0?0(h0, x, R) for fixed values of x
and R we individuate the saddle point (h0s, ?0s),
that satisfies condition ??0/ ? h0 0, by
selecting the eigenvalue with the largest
imaginary part, using multidimensional maps
R 35, x/D 4. Frequency and temporal growth
rate Level curves. ?0cost (thick curves),
r0cost (thin curves)
s0
k0
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?0(k0,s0) - R 35, x/D 4.
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r0(k0,s0) - R 35, x/D 4.
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Saddle points determination is very sensitive to
Orr-Sommerfeld boundary conditions and to number
and choice of collocation points for order zero
numerical resolution.
R 50, x/D 7. Frequency and temporal growth
rate Level curves. ?0cost (dashed curves),
r0cost (solid curves).
s0
This aspect becomes more relevant when y-domain
is getting larger, that is, for smaller R and
larger x values. For this reason, we use
truncated Laurent series to extrapolate saddle
points behavior in x from data at lower x values,
that are more accurate.
k0
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s0(x)
k0(x)
x
x
R 35 Saddle points (open circles) and
extrapolated curve (solid line)
Once known h0s(x) in this way, the relative
?0s(x) are given by dispersion relation. The
system is now perturbed, at every longitudinal
station, with those characteristics that at
order zero turn out to be locally the most
unstable (in absolute sense) for base flow.
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Im(?K1) Im(K2) R35
R50 R100
Re(?K1) Re(K2) R35
R50 R100
x
x
Coefficients (Real and Imaginary part) of
evolution equation for modulation

- R 35, 50, 100
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where is the adjoint eigenfunction
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Spatial growth rate
Wave number
k0 k R35 R50 R100
s0 s R35 R50 R100
Frequency
Temporal growth rate
r0 r R35 R50 R100
?0 ? R35 R50 R100
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Frequency. Comparison between the present
solution (R35,50,100), Zebib's numerical study
(1987), Piers direct numerical simulations
(2002), Williamson's experimental results (1988).
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Temporal growth rate. Comparison between the
present solution (R35,50) and Zebib's numerical
study (1987).
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Validity limits for the near-parallel flow First
order corrections are acceptable when they are
much lower than the corresponding order zero
values where they are not so, parallel flow
theory is no longer valid. A possible criterion
to establish this, is the following

where f is one of the stability
characteristics and ? is the wave length in
x. For fixed R values, these conditions are more
restrictive for temporal characteristics than for
the spatial ones. Spatial growth rate s seems to
be already well described at order zero, while
frequency ? is the characteristic which is more
influenced by first order corrections.
Increasing R, the region in which the flow can
no more be considered parallel becomes larger
this region involves not only the near but also
part of the intermediate wake.
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Conclusions Validity limits for parallel theory
by observing first order corrections, the flow
cannot be supposed parallel in the near wake and
also in a relevant portion of the intermediate
wake. System stability for what said about
acceptable first order corrections, the
intermediate and far wake is convectively
unstable. Positive temporal growth rate values
are considered not acceptable, even if they are
in a region of the domain (the beginning of near
wake) in which they would be experimentally
confirmed. Second order corrections (?2) seem
to be unnecessary, for they would not affect
results so much in the region where parallel flow
theory is valid and they would be completely
useless where first order corrections are already
too big.
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Inner and outer expansions Details up to third
order
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Order 0
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Order 1
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Order 2
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Order 3
C3 to be determined with boundary conditions in
xx
where